Question: Factor the following expression: $8$ $x^2$ $-41$ $x+$ $36$
Answer: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(8)}{(36)} &=& 288 \\ {a} + {b} &=& & & {-41} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $288$ and add them together. The factors that add up to ${-41}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-9}$ and ${b}$ is ${-32}$ $ \begin{eqnarray} {ab} &=& ({-9})({-32}) &=& 288 \\ {a} + {b} &=& {-9} + {-32} &=& -41 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {8}x^2 {-9}x {-32}x +{36} $ Group the terms so that there is a common factor in each group: $ ({8}x^2 {-9}x) + ({-32}x +{36}) $ Factor out the common factors: $ x(8x - 9) - 4(8x - 9) $ Notice how $(8x - 9)$ has become a common factor. Factor this out to find the answer. $(8x - 9)(x - 4)$